SH1-SH2 Confidence intervals for a mean from large samples Quiz: AQA Further Maths, Unit 4

The 20 questions

1. For a normal population with known variance sigma^2, which expression gives the symmetric 95% confidence interval for the population mean from a sample of n observations with mean xbar?

   • xbar ± 1.645 × sigma/sqrt(n)
   • xbar ± 1.96 × sigma^2/n
   • xbar ± 1.96 × sigma × sqrt(n)
   • xbar ± 1.96 × sigma/sqrt(n)

2. What is the standard error of the sample mean for n independent observations from a population with standard deviation sigma?

   • sqrt(sigma/n)
   • sigma^2/sqrt(n)
   • sigma/n
   • sigma/sqrt(n)

3. Which z value gives a symmetric 99% confidence interval for a normal mean, using the standard normal distribution?

   • 2.326
   • 1.645
   • 1.960
   • 2.576

4. Which z value gives a symmetric 90% confidence interval for a normal mean?

   • 1.282
   • 2.576
   • 1.645
   • 1.960

5. What does the confidence level of 95% mean for a symmetric confidence interval?

   • The sample mean lies inside the interval with probability 0.95 after each new sample is taken
   • There is a 95% probability that the true mean lies inside this particular calculated interval
   • About 95% of the individual data values lie inside the calculated interval
   • In repeated sampling, about 95% of intervals constructed this way would contain the true population mean

6. A sample of n = 36 is taken from a normal population with known sigma = 6. The sample mean is 50. What is the 95% confidence interval for the population mean?

   • (48.04, 51.96)
   • (48.04, 52.96)
   • (49.00, 51.00)
   • (44.12, 55.88)

7. A sample of n = 25 from a normal population with known sigma = 2 has mean 17.3. What is the symmetric 99% confidence interval for the population mean, to 2 decimal places?

   • (16.27, 18.33)
   • (15.97, 18.63)
   • (16.52, 18.08)
   • (17.30, 18.70)

8. A sample of size n is multiplied by four, with sigma and the confidence level unchanged. How does the width of the symmetric confidence interval for the mean change?

   • It halves, because the width is proportional to 1/sqrt(n)
   • It is unchanged, because the width depends only on sigma and the confidence level
   • It quadruples, because the width is proportional to n
   • It doubles, because the width is proportional to sqrt(n)

9. A normal population has known sigma = 4.5. What is the smallest sample size for a symmetric 95% confidence interval for the mean to have total width at most 2?

   • 39
   • 78
   • 156
   • 77

10. A sample of n = 100 from a population with unknown variance has mean 24.6 and standard deviation 5.2. Using the large-sample method, what is the approximate symmetric 95% confidence interval for the mean?

    • (14.41, 34.79)
    • (23.26, 25.94)
    • (23.58, 25.62)
    • (24.08, 25.12)

11. Why is the sample standard deviation s used in place of sigma when constructing a large-sample confidence interval for a mean?

    • The t-distribution must always be used instead of the normal distribution when sigma is unknown
    • s is always larger than sigma, so the interval it gives is wider and therefore safer
    • For large n, s is a good estimate of sigma, so the normal approximation with z values remains reasonable
    • The central limit theorem makes the sample standard deviation exactly equal to sigma for any sample

12. A symmetric 95% confidence interval for a population mean is (12.1, 15.9). What is the sample mean used to construct it?

    • 14.0
    • 3.8
    • 15.9
    • 13.0

13. A symmetric 95% confidence interval for a population mean is (12.1, 15.9). Assuming the z-based method, what is the standard error used?

    • Approximately 1.900
    • Approximately 0.950
    • Approximately 0.969
    • Approximately 3.724

14. Keeping the data fixed, how does a symmetric 95% confidence interval for a mean change when the confidence level is raised to 99%?

    • It moves to a new centre, because the sample mean changes with the confidence level
    • It becomes wider, because the z value rises from 1.96 to 2.576
    • It becomes narrower, because the z value falls from 1.96 to 1.645
    • It is unchanged, because the same sample data are used

15. A sample of n = 16 from a normal population with known sigma gives the 95% confidence interval (42.0, 48.0) for the mean. What is sigma, to 3 significant figures?

    • 3.06
    • 12.2
    • 1.53
    • 6.12

16. A normal population has known sigma = 3. What is the smallest sample size needed for a symmetric 95% confidence interval for the mean to have margin of error at most 0.5?

    • 138
    • 277
    • 139
    • 35

17. A symmetric 90% confidence interval is to be found for the mean of a normal population with known sigma = 10, using n = 25 observations with sample mean 60. What is the interval?

    • (56.08, 63.92)
    • (53.42, 66.58)
    • (58.00, 62.00)
    • (56.71, 63.29)

18. Which statement about a symmetric 95% confidence interval is correct?

    • Before the sample is taken, there is a 95% probability that the random interval will contain the true population mean
    • After the sample is taken, there is a 95% probability that the true mean lies within this specific interval
    • The sample mean will lie inside the calculated interval for 95% of future samples
    • 95% of the individual observations in the sample lie inside the calculated interval

19. A student has n = 12 normal observations with unknown variance. The student uses the sample standard deviation s in a z-based 95% interval. What is the best criticism of this method?

    • The z-based interval is too narrow, since the t-distribution with 11 degrees of freedom (critical value about 2.201) should be used
    • Using s always gives a narrower interval than using the unknown sigma would give
    • The critical value should be 2.576 because s is only a sample estimate of sigma
    • The z-based interval is valid because n = 12 is large enough for the central limit theorem to apply

20. A sample of n = 400 from a population with unknown variance has mean 50 and standard deviation 8. What is the approximate symmetric 95% confidence interval for the mean?

    • (48.43, 51.57)
    • (49.61, 50.39)
    • (42.16, 57.84)
    • (49.22, 50.78)

Related quizzes

• Discrete random variables and their measures Quiz · SA1-SA3 · 20 questions
• Expectation and linear functions Quiz · SA4-SA5 · 20 questions
• Discrete uniform distribution Quiz · SA6-SA7 · 20 questions
• Modelling with the Poisson distribution Quiz · SB1-SB2 · 20 questions
• Mean, variance and sums of Poisson distributions Quiz · SB3-SB4 · 20 questions
• Hypothesis test for a Poisson mean Quiz · SB5 · 20 questions
• Type I and Type II errors and the power of a test · SC1-SC2 · 20 questions
• Probability density functions, medians and quartiles Quiz · SD1-SD3 · 20 questions
• Mean, variance and linear functions of continuous variables Quiz · SD4-SD5 · 20 questions
• Cumulative distribution functions and the rectangular distribution Quiz · SD6-SD8 · 20 questions